3. G RAPHS AND CW- COMPLEXES
Graphs.
Definition. A directed graph X consists of two disjoint sets, V (X) and E(X)
(the set of vertices and edges of X, resp.), together with two mappings o,
t : E(X) → V (X).
Vertices o(e), t(e) are the endpoints of e,
Definition. A graph is a directed graph X together with a mapping E(X) →
E(X), e 7→ ē, satisfying: e 6= ē, ē¯ = e, o(ē) = t(e) and (consequently) t(ē) =
o(e), for all e ∈ E(X).
An unoriented edge of X is a pair {e, ē}, where e ∈ E(X).
Depict vertices by small circles, and edges e as line segments joining
their endpoints, with an arrow pointing from o(e) to t(e). Unoriented edges
are depicted as line segments without an arrow.
e
e
-
e
e
e
e
e
{e, ē}
ē
Note: o(e) = t(e) is allowed.
Definition. An orientation of a graph X is a set containing exactly one edge
from each unoriented edge {e, ē}.
Definition. Let X and Y be graphs. A graph map from X to Y is a mapping
f : V (X) ∪ E(X) → V (Y ) ∪ E(Y ) which maps vertices to vertices and edges
to edges, such that, for all edges e ∈ V (X), f (o(e)) = o( f (e)), f (t(e)) =
t( f (e)) and f (ē) = f (e).
Graph map f is called an isomorphism if it is bijective.
A graph X is a subgraph of a graph Y if V (X) ⊆ V (Y ), E(X) ⊆ E(Y )
and the inclusion map V (X) ∪ E(X) ,→ V (Y ) ∪ E(Y ) is a graph map, i.e. if
e ∈ E(X) then o(e), t(e) and ē have the same meaning in Y as they do in X.
Paths. Let Ln (n ≥ 0) be the graph with vertex set {0, 1, . . . , n} and edge set
{(0, 1), (1, 2), . . . , (n − 1, n)} ∪ {(1, 0), (2, 1), . . . , (n, n − 1)}
with o(i, i + 1) = i, t(i, i + 1) = i + 1, (i, i + 1) = (i + 1, i).
Ln :
0
e
1
e
2e
···
n −e1
ne
Definition. A path of length n in a graph X is a graph map p : Ln → X.
Denote p(0) by o(p) and p(n) by t(p) and call these the endpoints of p.
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2
[Say that p joins o(p) to t(p), and that it is a path from o(p) to t(p).]
A path p of length n > 0 can be viewed as a sequence of edges (e1 , . . . , en ),
where t(ei ) = o(ei+1 ) for 1 ≤ i ≤ n − 1, and o(p) = o(e1 ), t(p) = t(en )
(ei = p(i − 1, i)).
Also, for every vertex v of X, there is a trivial path, denoted by 1v , with
o(1v ) = t(1v ) = v, of length 0.
Definition. A graph X is connected if, given x, y ∈ V (X), there exists a path
in X from x to y.
A path (e1 , . . . , en ) is reduced if ei 6= ēi+1 for 1 ≤ i ≤ n − 1 (trivial paths are
reduced).
A path p is closed if o(p) = t(p).
A circuit is a closed reduced path of positive length, (e1 , . . . , en ) such that
o(ei ) 6= o(e j ) for 1 ≤ i, j ≤ n with i 6= j.
Definition. A tree is a connected graph having no circuits.
Lemma 3.1. A graph X is a tree ⇔ given vertices u, v of X, there is a
unique reduced path which joins u to v.
Proof. Omitted.
A subtree of a graph is a subgraph which is a tree.
A maximal tree in a graph X is a subtree of X which is not properly contained in any other subtree of X.
Lemma 3.2. Let X be a connected graph. Then a subtree T of X is maximal
if and only if V (T ) = V (X). Further, X has a maximal subtree.
Proof. Omitted.
Fundamental group. Given two paths p = (e1 , . . . , en ) and q = ( f1 , . . . , fm ),
with t(p) = o(q), the product pq is defined to be (e1 , . . . , en , f1 , . . . , fm ), a
path from o(p) to t(q). Note: if q is trivial, pq = p, and if p is trivial then
pq = q.
If p = (e1 , . . . , en ) is a path, p̄ means the path (ēn , . . . , ē1 ) (1̄v = 1v ).
Note: for any path p, p̄¯ = p, o( p̄) = t(p), t( p̄) = o(p).
Let p, q be paths; p ' q means one is obtained from the other by deleting a
pair of edges of the form eē.
Definition. Paths p and q are freely equivalent, written p ∼ q, if there is a
sequence of paths p = p1 , p2 , . . . pn = q, where pi ' pi+1 for 1 ≤ i ≤ n − 1.
Equivalence relation on the set of all paths; denote equivalence class of p
by [p].
Note: freely equivalent paths have the same endpoints.
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Lemma 3.3. Any path is equivalent to a unique reduced path.
Proof. Just like the proof of Lemma 1.2.
Lemma 3.4. (1) If p ∼ p0 , q ∼ q0 and t(p) = o(q) then pq ∼ p0 q0 ;
(2) If p ∼ q then p̄ ∼ q̄.
Proof. Exercise.
Let v0 be a vertex of the connected graph X. Let π1 (X, v0 ) be the set of
equivalence classes of paths starting and ending at v0 .
If [p], [q] ∈ π1 (X, v0 ), then by Lemma 3.4(1), can define their product by
[p][q] = [pq]. This makes π1 (X, v0 ) into a group.
Identity elt is [1v0 ], and [p]−1 is [ p̄].
Definition. The group π1 (X, v0 ) is called the fundamental group of X at v0 .
Now choose a maximal tree T of X. For v ∈ V (X), let pv be the reduced
path in T from v0 to v. For e ∈ E(X), let we = po(e) ept(e) . Then [we ] ∈
π1 (X, v0 ), and [we ] = 1 if e ∈ E(T ). If e 6∈ E(T ) then we is reduced. Further,
wē = po(ē) ē pt(ē) = pt(e) ē po(e) = we
hence [we ]−1 = [wē ].
Suppose p = e1 . . . en is a path from v0 to v0 in X. Then we1 . . . wen ∼ p, so
[p] = [we1 ] . . . [wen ].
Let A be an orientation of X.
Follows: π1 (X, v0 ) is generated by U = {[we ] | e ∈ A \ E(T )}.
Theorem 3.5. In fact, π1 (X, v0 ) is free with basis U.
Proof. Omitted.
Cayley Graphs. Let G be a gp, α : X → G a mapping such that α(X)
generates G. Form a directed graph:
set of vertices is G;
set of edges is G × X; edge (g, x) has label x.
o(g, x) = g, t(g, x) = gα(x).
For every edge e = (g, x) add an edge ē, with label x−1 , defining o(ē) =
gα(x), t(ē) = g.
Get a graph, the Cayley graph of G wrt α, denoted Γ(G, α).
Usually, α is suppressed and graph is denoted by Γ(G, X).
CW-complexes.
Let
E n = {x ∈ Rn | kxk ≤ 1} ,
U n = {x ∈ Rn | kxk < 1} ,
Sn−1 = E n −U n .
Let A be a Hausdorff space, Λ a set; for λ ∈ Λ, let Eλ be a copy of E n , so
containing a copy Sλ , Uλ of Sn−1 , U n . Let fλ : Sλ → A be a cts map, for
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`
λ ∈ Λ. Let Z = A q λ ∈Λ Eλ , and let X be the quotient space Z/ ∼, where
∼ is the equiv. rel. gend by z ∼ fλ (z), ∀z ∈ Sλ , ∀λ ∈ Λ. Let q : Z → X be
the quotient map: q maps A homeomorphically onto a closed subspace of
X, which is identified with A via q. Also, X is Hausdorff.
Definition. Let Y be a Hausdorff space having A as a subspace. Then Y is
obtained from A by adjoining n-cells if ∃ a homeom. h : X → Y , for some
X constructed as above, with h|A = idA .
Put p = h ◦ q, let cλ = p(Eλ ) (the n-cells), let pλ = p|Eλ (the characteristic
maps), and let fλ = pλ |Sλ (the attaching maps).
Facts:
(i) A is closed in Y .
◦
(ii) pλ |Uλ is a homeom. onto an open subset cλ of Y whose closure in Y is
cλ .
◦
(iii)The cλ are the path components of Y − A.
(iv) A subset B of Y is closed iff B ∩ A is closed in A and B ∩ cλ is closed in
cλ , for all λ ∈ Λ.
Definition. A top. space X is a CW-complex if ∃ subspaces
X 0 ⊆ X 1 ⊂ X 2 ⊆ . . . with X =
[
Xn
n≥0
such that each X n is closed, X 0 is discrete, X n is obtained from X n−1 (n ≥ 1)
by adjoining n-cells and a subset Y of X is closed iff Y ∩ X n is closed in X n ,
for all n ≥ 0.
Note: E 1 = [−1, 1], U 1 = (−1, 1).
Let X be a 1-dimensional CW-complex (X = X 1 ), let c be a 1-cell. If g1 , g2
◦
are homeoms. U 1 → c, define g1 ∼ g2 to mean g−1
1 g2 is strictly increasing.
(Equiv. rel. on set of such homeoms. with two equivalence classes, [g] and
[g0 ], where g0 (t) = g(−t).)
An oriented 1-cell is a pair (c, [g]) where c is a 1-cell and [g] is one of the
equivalence classes. Define (c, [g]) = (c, [g0 ]). If ϕ : [−1, 1] → c is the char.
map, put g = ϕ|(−1,1) . Then define o(c, [g]) = ϕ(−1), t(c, [g]) = ϕ(1). This
defines a graph Γ, with V (Γ) = X 0 , E(Γ) the set of oriented 1-cells.
If c is a 1-cell with char. map ϕ and g = ϕ|(0,1) , e = (c, [g]), then there
is a path αe in X from o(e) to t(e), αe = ϕ ◦ h, where h : [0, 1] → [−1, 1]
is h(t) = 2t − 1. Define αē to be ϕ 0 ◦ h (where ϕ 0 (t) = ϕ(−t)), so αē (t) =
αe (1 − t), hence αe αē is homotopic to the constant path at o(e).
If p = (e1 , . . . , em ) is a path in Γ, define
θ (p) = homotopy class of αe1 . . . αem
(bracketing is unimportant)
and let θ (1v ) =homotopy class of const. path at v, v ∈ V (Γ).
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Follows: if p ∼ q then θ (p) = θ (q). Hence, if v ∈ V (Γ), ∃ an induced
map ψ : π1 (Γ, v) → π1 (X, v), [p] 7→ θ (p).
Theorem 3.6. The map ψ : π1 (Γ, v) → π1 (X, v), is a group isomorphism,
so π1 (X, v) is a free group.
Proof. See Massey’s book, Thm 5.1, Ch.6 (needs Seifert-van Kampen Theorem).
2-complexes. Let X be obtained from A by attaching 2-cells. Let {cλ | λ ∈ Λ}
be the set of 2-cells, with char. maps pλ . Assume A path connected.
Let g : [0, 1] → S1 be the map t 7→ e2πit . Then αλ := pλ ◦ g is a closed
path in A. Choose v ∈ A and a path βλ in A from v to αλ (0). Then βλ αλ βλ−1
is a closed path at v, so its homotopy class wλ is in π1 (A, v).
Theorem 3.7. The homomorphism π1 (A, v) → π1 (X, v) induced by the inclusion map is onto, and the kernel is the normal subgroup of π1 (A, v) generated by {wλ | λ ∈ Λ}.
Proof. See Massey, Thm 2.1, Ch.7 (another application of Seifert-van Kampen).
Let hX | Ri be a group presn. Let K 0 = {v}, and let K 1 be obtained from
K 0 by adjoining a set of 1-cells {cx | x ∈ X}. (Only one choice for attaching
maps.)
[“Bouquet of circles” joined at a single point.]
Let px be the char. map, gx its restriction to (−1, 1), ex = (cx , gx ) corr.
oriented edge, and define ex−1 = ēx . For oriented edge ey , y ∈ X ±1 , let
αy = αey be the corr. path in K 1 as above. (preceding Thm 3.6).
[If y = x±1 , x ∈ X, αy goes once round cx in a preferred direction.]
Let F be the free gp on X. By Thms 3.5 and 3.6, x 7→ homotopy class of
αx extends to an isomorphism Φ : F → π1 (K 1 , x).
If r ∈ R, say r = y1 . . . yk , yi ∈ X ±1 , define αr = αy1 . . . αyk . Then αr
induces βr : S1 → K 1 with αr = βr ◦ g (g : t 7→ e2πit ).
Adjoin 2-cells {cr | r ∈ R} to K 1 with βr as attaching map for cr , to obtain
a CW-cx K = K(X | R). By Thm 3.7, π1 (K, v) ∼
= π1 (K 1 , v)/N, N being the
normal subgp gend by {wr | r ∈ R}, where wr is the htopy class of αr , ie
wr = Φ(r).
Hence π1 (K, x) ∼
= F/N 0 , where N 0 is the normal subgp of F gend by R,
so π1 (K, x) has presn hX | Ri.
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Examples. (1) hx, y | xyx−1 y−1 i; K ≈ the torus S1 × S1 .
c
x
c
y
(2) hx | x2 i; K is the real projective plane RP2 , obtained by identifying
antipodal points of S2 .
(3) hx, y | xyx−1 yi; K is the Klein bottle. Presn transforms to ha, b | a2 = b2 i
by Tietze transfs.